The 20th-ranked Blue Devils are playing No. 1 Florida State on Saturday for an ACC championship and an Orange Bowl berth.

Still, it wasn’t that long ago when Duke defensive tackle Sydney Sarmiento heard somebody wondering why the perpetually putrid program even bothered to field a football team.

“Not the butt of everybody’s joke anymore,” Sarmiento said Tuesday.

The Blue Devils have made believers out of the Seminoles, who routed Duke 48-7 last year in Tallahassee.

“Just as Florida State has bought into coach (Jimbo) Fisher’s coaching, I think those guys have bought into their team,” Seminoles safety Lamarcus Joyner said. “You just see a group of guys that believe in each other, and when you have a belief you can do a lot of wonders. ... I just see a team that believes in each other and has a great coach and some smart players.

“They’re on a roll.”

Now they have to find a way to beat a Seminoles team that’s a 29-point favorite and on track to play for the BCS championship. They also must figure out how to deal with quarterback Jameis Winston — the ACC’s rookie of the year, a leading Heisman Trophy candidate and the subject of an ongoing investigation into sexual assault allegations.

“Our entire season has been based on people telling us what we can’t do,” tight end Braxton Deaver said. “‘You’re last in the ACC’ — oh, that’s funny, now we’re first in the Coastal (Division).”

But what looks from the outside like an out-of-nowhere rise actually has been the product of coach David Cutcliffe’s patient reconstruction project.

“It’s not an accident. It’s a process,” Cutcliffe said. “Those young men on the 2008 team, ’09 team, ’10 team all took part in the development of these seniors and upperclassmen. ... It’s not just a year.

“It’s not just an event that happened this year.”

Picked in the preseason to finish last in the division, Duke (10-2, 6-2) has already set a school record for victories, captured its first division title and sewn up the first consecutive bowl berths in school history.

That’s quite a turnaround for a program that went winless four times from 1996-2007 with two other one-win seasons while setting the standard for football futility among power-conference schools during what defensive end Kenny Anunike called “the dark years.”

Deaver said when he arrived in 2010, some long-gone veterans from the previous regime carried an attitude of “I’m going to get my degree and get out of here.” And according to Anunike, some of those ex-teammates didn’t believe they could win games and instead went into them hoping merely to keep it close.

“We knew football was on the low end” of the totem pole, said running back Josh Snead, a Durham native. “We knew it was going to take time. Coach Cut, when he recruited me, told me, ‘Just give me a few years and you’ll see the changes.’”

Indeed, this Duke program is completely different from the one Cutcliffe inherited from Ted Roof.

These Blue Devils were built in the weight room — one of Cutcliffe’s first goals upon his hiring was having the players lose a combined 1,000 pounds during offseason conditioning — and developed through a patient redshirting process.

Long gone are those days when Duke was pressed into playing most of its freshmen before they were ready. Of the 52 players on the offensive and defensive depth charts, 34 have been redshirted at least once — including each of the 10 offensive linemen, all three tight ends, and eight of the 10 along the defensive line.

Only four player on the depth chart are in their first year in the program. Conversely, every offensive lineman is a redshirt sophomore or older — and the starters along both lines have stayed healthy enough to start every game.

“That’s basically loading your back end,” Anunike said. “You might sacrifice a few opportunities where guys might be able to play, but (Cutcliffe is) looking at what they’ll blossom into later in their redshirt junior year, redshirt senior year, and see how much more they’ll benefit the program than if they got thrown right into the fire.”

That patience paid off in a most unlikely division title — and a chance at an even bigger accomplishment this weekend in Charlotte. Fisher says the Blue Devils caught his eye as a possible opponent about three weeks ago.

“You see highlights and you see the numbers and what’s going on,” Fisher said. “I knew if they got hot, they could be there.”

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